Using the ‘BayesFactor’ package, version 0.9.2+

Functions to manipulate Bayes factor objects

The t test section below has examples showing how to manipulate Bayes factor objects, but all these functions will work with Bayes factors generated from any function in the BayesFactor package.

One-sample tests (and paired)

We use the sleep data set in R to demonstrate a one-sample t test. This is a paired design; for details about the data set, see ?sleep. One way of analyzing these data is to compute difference scores by subtracting a participant’s score in one condition from their score in the other:

data(sleep)

## Compute difference scores
diffScores = sleep$extra[1:10] - sleep$extra[11:20]

## Traditional two-tailed t test
t.test(diffScores)

## 
##  One Sample t-test
## 
## data:  diffScores
## t = -4, df = 9, p-value = 0.003
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -2.46 -0.70
## sample estimates:
## mean of x 
##     -1.58

We can do a Bayesian version of this analysis using the ttestBF function, which performs the “JZS” t test described by Rouder, Speckman, Sun, Morey, and Iverson (2009). In this model, the true standardized difference \(\delta=(\mu-\mu_0)/\sigma_\epsilon\) is assumed to be 0 under the null hypothesis, and \(\text{Cauchy}(\text{scale}=r)\) under the alternative. The default \(r\) scale in BayesFactor for t tests is \(\sqrt{2}/2\). See ?ttestBF for more details.

bf = ttestBF(x = diffScores)
## Equivalently:
## bf = ttestBF(x = sleep$extra[1:10],y=sleep$extra[11:20], paired=TRUE)
bf

## Bayes factor analysis
## --------------
## [1] Alt., r=0.707 : 17.3 ±0%
## 
## Against denominator:
##   Null, mu = 0 
## ---
## Bayes factor type: BFoneSample, JZS

The bf object contains the Bayes factor, and shows the numerator and denominator models for the Bayes factor comparison. In our case, the Bayes factor for the comparison of the alternative versus the null is 17.259. After the Bayes factor is a proportional error estimate on the Bayes factor.

There are a number of operations we can perform on our Bayes factor, such as taking the reciprocal:

1 / bf

## Bayes factor analysis
## --------------
## [1] Null, mu=0 : 0.0579 ±0%
## 
## Against denominator:
##   Alternative, r = 0.707106781186548, mu =/= 0 
## ---
## Bayes factor type: BFoneSample, JZS

or sampling from the posterior of the numerator model:

chains = posterior(bf, iterations = 1000)
summary(chains)

## 
## Iterations = 1:1000
## Thinning interval = 1 
## Number of chains = 1 
## Sample size per chain = 1000 
## 
## 1. Empirical mean and standard deviation for each variable,
##    plus standard error of the mean:
## 
##        Mean     SD Naive SE Time-series SE
## mu    -1.42  0.436   0.0138         0.0154
## sig2   2.02  1.157   0.0366         0.0395
## delta -1.11  0.427   0.0135         0.0162
## g      6.26 58.623   1.8538         1.8538
## 
## 2. Quantiles for each variable:
## 
##         2.5%    25%   50%    75%  97.5%
## mu    -2.289 -1.705 -1.43 -1.141 -0.597
## sig2   0.744  1.270  1.69  2.446  5.223
## delta -1.973 -1.383 -1.08 -0.813 -0.347
## g      0.176  0.592  1.13  2.928 33.734

The posterior function returns a object of type BFmcmc, which inherits the methods of the mcmc class from the coda package. We can thus use summary, plot, and other useful methods on the result of posterior. If we were unhappy with the number of iterations we sampled for chains, we can recompute with more iterations, and then plot the results:

chains2 = recompute(chains, iterations = 10000)
plot(chains2[,1:2])

Directional hypotheses can also be tested with ttestBF (Morey & Rouder, 2011). The argument nullInterval can be passed as a vector of length 2, and defines an interval to compare to the point null. If null interval is defined, two Bayes factors are returned: the Bayes factor of the null interval against the alternative, and the Bayes factor of the complement of the interval to the point null.

Suppose, for instance, we wanted to test the one-sided hypotheses that \(\delta<0\) versus the point null. We set nullInterval to c(-Inf,0):

bfInterval = ttestBF(x = diffScores, nullInterval=c(-Inf,0))
bfInterval

## Bayes factor analysis
## --------------
## [1] Alt., r=0.707 -Inf<d<0    : 34.4  ±0%
## [2] Alt., r=0.707 !(-Inf<d<0) : 0.101 ±0%
## 
## Against denominator:
##   Null, mu = 0 
## ---
## Bayes factor type: BFoneSample, JZS

We may not be interested in tests against the point null. If we are interested in the Bayes factor test that \(\delta<0\) versus \(\delta>0\) we can compute it using the result above. Since the object contains two Bayes factors, both with the same denominator, and \[ \left.\frac{A}{C}\middle/\frac{B}{C}\right. = \frac{A}{B}, \] we can divide the two Bayes factors in bfInferval to obtain the desired test:

bfInterval[1] / bfInterval[2]

## Bayes factor analysis
## --------------
## [1] Alt., r=0.707 -Inf<d<0 : 341 ±0%
## 
## Against denominator:
##   Alternative, r = 0.707106781186548, mu =/= 0 !(-Inf<d<0) 
## ---
## Bayes factor type: BFoneSample, JZS

The Bayes factor is about 340.

When we have multiple Bayes factors that all have the same denominator, we can concatenate them into one object using the c function. Since bf and bfInterval both share the point null denominator, we can do this:

allbf = c(bf, bfInterval)
allbf

## Bayes factor analysis
## --------------
## [1] Alt., r=0.707             : 17.3  ±0%
## [2] Alt., r=0.707 -Inf<d<0    : 34.4  ±0%
## [3] Alt., r=0.707 !(-Inf<d<0) : 0.101 ±0%
## 
## Against denominator:
##   Null, mu = 0 
## ---
## Bayes factor type: BFoneSample, JZS

The object allbf now contains three Bayes factors, all of which share the same denominator. If you try to concatenate Bayes factors that do not share the same denominator, BayesFactor will return an error.

When you have a Bayes factor object with several numerators, there are several interesting ways to manipulate them. For instance, we can plot the Bayes factor object to obtain a graphical representation of the Bayes factors:

plot(allbf)

We can also divide a Bayes factor object by itself — or by a subset of itself — to obtain pairwise comparisons:

bfmat = allbf / allbf
bfmat

##                            denominator
## numerator                   Alt., r=0.707 Alt., r=0.707 -Inf<d<0
##   Alt., r=0.707                   1.00000                0.50146
##   Alt., r=0.707 -Inf<d<0          1.99416                1.00000
##   Alt., r=0.707 !(-Inf<d<0)       0.00584                0.00293
##                            denominator
## numerator                   Alt., r=0.707 !(-Inf<d<0)
##   Alt., r=0.707                                   171
##   Alt., r=0.707 -Inf<d<0                          341
##   Alt., r=0.707 !(-Inf<d<0)                         1

The resulting object is of type BFBayesFactorList, and is a list of Bayes factor comparisons all of the same numerators compared to different denominators. The resulting matrix can be subsetted to return individual Bayes factor objects, or new BFBayesFactorLists:

bfmat[,2]

## Bayes factor analysis
## --------------
## [1] Alt., r=0.707             : 0.501   ±0%
## [2] Alt., r=0.707 -Inf<d<0    : 1       ±0%
## [3] Alt., r=0.707 !(-Inf<d<0) : 0.00293 ±0%
## 
## Against denominator:
##   Alternative, r = 0.707106781186548, mu =/= 0 -Inf<d<0 
## ---
## Bayes factor type: BFoneSample, JZS

bfmat[1,]

##                denominator
## numerator       Alt., r=0.707 Alt., r=0.707 -Inf<d<0 Alt., r=0.707 !(-Inf<d<0)
##   Alt., r=0.707             1                  0.501                       171

and they can also be transposed:

bfmat[,1:2]

##                            denominator
## numerator                   Alt., r=0.707 Alt., r=0.707 -Inf<d<0
##   Alt., r=0.707                   1.00000                0.50146
##   Alt., r=0.707 -Inf<d<0          1.99416                1.00000
##   Alt., r=0.707 !(-Inf<d<0)       0.00584                0.00293

t(bfmat[,1:2])

##                         denominator
## numerator                Alt., r=0.707 Alt., r=0.707 -Inf<d<0
##   Alt., r=0.707                   1.00                  0.501
##   Alt., r=0.707 -Inf<d<0          1.99                  1.000
##                         denominator
## numerator                Alt., r=0.707 !(-Inf<d<0)
##   Alt., r=0.707                                171
##   Alt., r=0.707 -Inf<d<0                       341

If these values are desired in matrix form, the as.matrix function can be used to obtain a matrix.

Two-sample test (independent groups)

The ttestBF function can also be used to compute Bayes factors in the two sample case as well. We use the chickwts data set to demonstrate the two-sample t test. The chickwts data set has six groups, but we reduce it to two for the demonstration.

data(chickwts)

## Restrict to two groups
chickwts = chickwts[chickwts$feed %in% c("horsebean","linseed"),]
## Drop unused factor levels
chickwts$feed = factor(chickwts$feed)

## Plot data
plot(weight ~  feed, data = chickwts, main = "Chick weights")